REQUIREMENTS FOR A VISUALIZATION SOFTWARE SYSTEM FOR 2010 Xah Lee, 2007-03-16
In this essay, i give a list of requirements that i think is necessary for a software system for creating scientific visualization for the next decade (2007-2017). (for a HTML version with images, please see http://xahlee.org/3d/viz.html ) For the past 10 years, i have been strongly interested in mathematical visualization. I'm not a professor, and am not doing it for the educational purposes per se. Geometry has just been beautiful for me, and i'm also a professional programer. Programing computers to visualize geometric theorems or geometric objects, has been a immense personal pleasure. A particular aspect of visualization, is to design it so that when viewed, it forces a immediate and crystal clear understanding of the object or theorem, to its viewers. I have for the past couple years sought for a software platform/system where i can build 3D objects, with abilities to do interactive manipulation (such as moving a slider to change a parametric surface's parameter), dynamic rotation (viewing from different angles instantaneously), animations (such as morphing that shows geometric processes), and most of all, walk-thru in it as if it is a building (e.g. the perception of actually walking on a moebius strip↗ or Klein bottle↗ is vastly different than just viewing it.). -------- Requirement: Real-Time 3D Rotation For math visualization, it is absolutely necessary to be able to rotate the object in real-time. For example, when viewing a complex surface, a molecule structure, or the design of a complex device. This requirement is so basic it warrants no further explanation. above: The Chen Gackstatter minimal surface as shown by the software 3D-XplorMath ↗. This surface has several 3-fold symmetries, but a static image does not show this. (Live Rotation with Java Applet) -------- Requirement: Real-time Interactivity The programer should be able to build buttons and sliders that change parameters and have the object reflect these changes visually right away. This real-time interactivity is frequently needed. For example, most math visualizations involve a function, a process, or a geometric object, that varies by parameters, and the key to the visualization is showing how the parameter effect the graph or object. Some Examples follows. Example: Moebius Transformation above: A software that plots a complex function (the Moebius transformation↗), by showing the pre-image and post-image of the function. The function is keyed by various parameters that can be represented as points in the plane. Dragging these points will change the image interactively. Source: MoeApplet by Paul Murray ↗ Example: Regular Polytopes above: a Java applet showing a slice of 4-dimensional symmetric polytopes↗ , with sliders that change the polytope's parameters. Dragging the polytope will rotate it in real-time. Hyperstar Java Applet by Mark Newbold ↗. Example: Plane Geometry above: A interactive plane geometry software showing a Equiangular Spiral and its inverse curve , with respect to a point O. Equiangular spiral is a family of curves varying by a parameter θ, from 0 to π/2. This parameter can be changed by dragging the point E of the angle DAE, and the corresponding curve will be shown (blue). Its inverse curve with respect to the point O is shown in pink. Dragging the point O will change the position of O, and corresponding inverse curve will be shown in real-time. above: A plane geometry software demonstrating Bezier curve. User can drag the poins to very the input, and see the polynomial change shape in real-time. Java Applet version. Example: Conic Sections Computing power today are some 50 times faster than a decade ago. However, math visualizations, have hardly advanced. For example, let's consider the simple task of plotting of polynomial equations. As we know, 2nd degree polynomials of 2 variables are conic sections. It would nice, that the program can show the general form A x^2 + B x y + C y^2 + D x + E y + F == 0 and let users change the constants by sliders, and have the graph change on the fly. Such a application would give immense feedback that reinforces learning, which is what visualization is about. Similiarly, this can be done for 3rd degree polynomials, and the visualization of polynomials of 3 variables. There are tens, if not hundreds, elementary equation and function graphing software. However, it is almost impossible to find one that shows the power of interactivity described above. Apple Computer Inc, bundled with OS X as current as of March 2007, a plotting software (called Grapher↗) that has zero interactivity features. And Mathematica's abilities for plotting polynomial equations in as late as 2005, is basically the same as 1996. Its ability for visual experiment and learning of polynomials of 3 variables is practically zero. I have personally played with perhaps over 20 major plotting software over the past 15 years, and i only know one that has this feature. (see Nucalc↗) -------- Requirement: easy-to-create, real-time animation The software system must have the capabilities for programers to build animations, and view the animation interactively in real-time (as opposed to generating the animation once and save it as a movie). For example, once a programer has coded the animation, she should easily play it forward and backward in the program, as well as changing parameters and re-run the new animation without going thru some compilation or export process. For example, if the animation involves a 3D object, the system should also allow the user to rotate it while the animation is on-going, all in real-time. For example, this system should allow programer to easily create a animation that morphs a sphere into a cube, or a bunch of spheres flying in space as a swarm of flies or in some artificial formation like a 3D-kaleidoscope. Example: Cycloidal Curves above: A movie generated by a math software, showing how the cycloid curve is generated by rolling a circle. (cycloidGen.mov). In this example, you can see the trace by a point on the boundary of the circle. What happens if the tracing point is nearer to the center of the circle? This is a example where a interactive slider will help. As a user adjust the slider, the animation should show its effects in real-time. Example: Cellular Automaton above: A software that computes the cellular automaton↗known as Game Of Life, showing it as a animation as it evolves. At any point in time, a user can add more cells to it by clicking in the window, and continue the animation to see its effects. Example: Helicoid-catenoid Surface Family above: A frame of a software generated animation, showing the isometry property of the surface family helicoid-catenoid. (helicoid- catenoid.mov) This is a example where it shows the importance of being able to rotate the object in real-time in a on-going animation. Example: Sphere-eversion above: The classic Sphere-eversion visualization problem, visually demonstrating how a sphere can be turned inside-out without creating any crease. (Source: John Sullivan et al. ↗. image source ↗, accessed on 2007-03-18.) (For detail about this problem, see: Smale's paradox↗) Examples of system that do NOT satisfy this requirement would be: Mathematica (up to version 5), 3D-XplorMath, most CAD systems. Most of these systems allow animations, but usually for a generate-once and save-as-static-movie-file, not designed for interactive view. Example: Stereographic Projection above: A illustration showing stereographic projection. (source) To create visualization for stereographic projection, ideally, the user should be able to rotate the whole plane, or move it up and down in relation to the sphere, as to see the effect on the mapping. Also, the user should be able to rotate the sphere and see the mapping change on the plane. The user should also be able to move a single projection line to see mappings of arbitrary points. Lastly, it is ideal in this visualization to let users change the pattern on plane or on the sphere to arbitrary images. For example, the plane should have a default set containing rectangular grids and polar grids. (showing the important property of circles mapping to circles/lines) The sphere should have standard projection of platonic solids. (showing the important concept of platonic solids as networks) And, the user should be able to put the graph of arbitrary complex function on the plane or a Riemann sphere↗ on the sphere. Today, it is non-trivial to create a visualization such as this. For example, the above image is created in Mathematica. One has to write a function that plots riemann sphere in the first place, to create the image of the sphere. The system also needs to have animation and interactive slider abilities, as to be able to make the sphere and plane rotate or reposition independently in real-time, and the moving of the projection line. Examples of visualization systems that do satisfy the animation requirement to various degrees discussed in this section would be: Macromedia Flash, VPython, 3D Game Engines. -------- Requirement: Walk-thru The software system should have the capability of walk-thru or fly- thru. That is, allow the programer precise control of the camera position and orientation in real-time. The most obvious application of this requirement is in CAD systems that are specialized for architecture. For example, if i build a fancy 3D-maze or elaborate cathedral, i should be able to say, walk inside it, so as to view the object from inside and as well as getting the real experience of perceiving such a building. The ability to do walk-thrus has significant applications in math visualization too, however, it is seldom realized. Here are some examples of its application. Example: Walking on Surfaces above: A hyperbolic-paraboloid, with a human character standing on top of it. (from the computer graphics virtual world Second Life. Credit: Henry Segerman↗, aka Seifert Surface.) Compare the above image with the following abstract rendering that is commonly found in math visualization software. above: a abstract rendering of the hyperbolic-paraboloid. Java Applet Perhaps due to the evolution of our perception system, the experience of actually walking on or inside a math object, is vastly different than just seeing it as zero-thickness abstract lines and sheets rotating around its center. Although, the virtual world created by computer graphics only fake the experience of actually being there, still, such a virtual world experience throws a impact on understanding. Imagine, demonstrating the one-sidedness property of surfaces such as the moebius strip↗ or Klein bottle↗ thru a live rotation and explanation, compared to having the student inadvertently taking a walk inside such a object and suddenly felt the single-sided oddity. A person experiencing such a situation will probably remember it as part of his life's memory. Example: 3D-Lattices The ability to do walk-thrus also fits well in visualizations involving 3-dimensional lattices, such as Crystallographic group↗, 3D- tilings (sphere packing), or visualizing periodic minimal surfaces. For example, the following images of periodic minimal surfaces are shown in the way they are traditionally presented, as a few representative patchs of the periodic surface, and users can rotate the surfaces in real-time: above: Periodic minimal surfaces↗. Source: Richard Palais, Karcher Hermann. ↗) As they are shown, their periodic structure is not visually clear, especially those non-rectangular ones. (e.g. the last 3) It also takes some effort to imagine how they actually fill the space. Now, imagine these shown in a walk-thru system, where a portion of space is filled with its repetitions, and the user effectively becomes a bee, flying about in it, able to observe detail in a close-up on any point of the surface from any angle, or getting a bird's eye's overview of the surface's lattice symmetry. Example: 3-Manifolds above: Screenshot of Jeff Week's Curved Space ↗ software. This program lets users fly-thru various 3-manifolds↗ . There is basically no other way of visualizing 3-manifolds other than the techniques of walk-thru. There is effectively no visualization system that lets programers easily code interactive walk-thrus. The possible exception is 3D Game Engines↗, however, it requires a specialized programer with years of dedicated experience to create any thing in it, and the work is rather tedious. Some such game engines have higher level (scripting) languages that let the user (players) create objects or scenes in- world. (A example is the Linden Scripting Language↗ of Second Life↗. ) -------- Requirement: Easy To Use For Average Programers The system must be easy to use for average programers or scientists. For example, a person who had just a year of programing experience, should be able to code a variety of visualization projects with the system. Often, these high-level languages are called scripting languages. Mathematica↗, Lisps (Logo↗, Scheme↗, Tcl/tk↗ ), Visual Basic↗, VPython↗, Macromedia Flash↗, CAD systems (AutoLISP↗ ), Linden Scripting Language↗ are examples of high-level systems. On the other hand, low-level language such as Java↗, C sharp↗, or, systems such as OpenGL↗ and Game Engines↗, or application programing systems such as Cocoa (API)↗ and X Window System↗, all require professional programers with years of experience to create a non- trivial application. These are not useful to most scientists or engineers. -------- Requirement: A Set of Basic Geometric Shapes The system must have a set of basic primitives or shapes that serve as building blocks. For example, spheres, torus, cubes, as well as meshes of triangles. Of course, the system should also have the following common elements: various coordinates systems, projection systems, lighting and camera control, build-in manifold by parametric surfaces and algebraic equations, and a whole set of 2D and 3D transformation functions (which can also work on n-dimensional coordinates). For a system aimed for math visualization, it would also be necessary to have access to a full range of math functions, such as integration, derivative, higher-order functions. [Continues on page 2] ----- This post is archived at I'm still working on the second part. Your opinions and comments on existing systems are welcome. Thanks. Xah [EMAIL PROTECTED] ∑ http://xahlee.org/ -- http://mail.python.org/mailman/listinfo/python-list
